Let a circle and a point not coinciding with its center C be given (Fig. 205). Three cases are possible: the point lies inside the circle (Fig. 205, a), on the circle (Fig. 205, b), outside the circle (Fig. 205, c). Let's draw a straight line that will intersect the circle at points K and L (in case b) the point will coincide with one of which will be closest to the point compared to all other points of the circle), and the other will be the most distant.
So, for example, in Fig. 205, and point K of the circle is closest to . In fact, for any other point on the circle, the broken line is longer than the segment SAG: but also therefore, on the contrary, for point L we find (again the broken line is longer than the straight line segment). We leave the analysis of the remaining two cases to the reader. Note that the largest distance is equal to the smallest if or if.
Let's move on to analyzing possible cases of arrangement of two circles (Fig. 206).
a) The centers of the circles coincide (Fig. 206, a). Such circles are called concentric. If the radii of these circles are not equal, then one of them lies inside the other. If the radii are equal, they coincide.
b) Let now the centers of the circles be different. Let's connect them with a straight line, it is called the line of centers of a given pair of circles. Mutual arrangement circles will depend only on the relationship between the value of the segment d connecting their centers and the values of the radii of the circles R, r. All possible significantly different cases are presented in Fig. 206 (counting).
1. The distance between centers is less than the difference in radii:
(Fig. 206, b), the small circle lies inside the large one. This also includes the case of a) coincidence of centers (d = 0).
2. The distance between centers is equal to the difference in radii:
(Fig. 206, s). The small circle lies inside the large one, but has one common point with it on the line of centers (they say that there is an internal tangency).
3. The distance between the centers is greater than the difference in radii, but less than their sum:
(Fig. 206, d). Each circle lies partly inside and partly outside the other.
The circles have two intersection points K and L, located symmetrically relative to the line of centers. A segment is a common chord of two intersecting circles. It is perpendicular to the line of centers.
4. The distance between centers is equal to the sum of the radii:
(Fig. 206, d). Each of the circles lies outside the other, but they have a common point on the line of centers (external tangency).
5. The distance between centers is greater than the sum of the radii: (Fig. 206, f). Each circle lies entirely outside the other. Circles have no common points.
The above classification follows completely from what has been discussed. above the question of the greatest and smallest distance from a point to a circle. You just need to consider two points on one of the circles: the closest and the farthest from the center of the second circle. For example, let's look at the case By condition. But the point of the small circle that is most distant from O is located at a distance from the center O. Therefore, the entire small circle lies inside the large circle. Other cases are considered in the same way.
In particular, if the radii of the circles are equal, then only the last three cases are possible: intersection, external tangency, external location.
Let the circles be given by a vector from the origin to the center and the radius of this circle.
Consider circles A and B with radii Ra and Rb and radius vectors (vector to the center) a and b. Moreover, Oa and Ob are their centers. Without loss of generality, we will assume that Ra > Rb.
Then the following conditions are satisfied:
Objective 1: Mansions of important noblesPoints of intersection of two circles
Suppose A and B intersect at two points. Let's find these intersection points.
To do this, a vector from a to a point P, which lies on the circle A and lies on OaOb. To do this, you need to take the vector b - a, which will be the vector between the two centers, normalize it (replace it with a codirectional unit vector) and multiply it by Ra. We denote the resulting vector as p. This configuration can be seen in Fig. 6
Rice. 6. Vectors a,b,p and where they live.
Let us denote i1 and i2 as vectors from a to the intersection points I1 and I2 of two circles. It becomes obvious that i1 and i2 are obtained by rotation from p. Because we know all the sides of the triangles OaI1Ob and OaI2Ob (Radius and distance between centers), we can get this angle fi, rotating the vector p in one direction will give I1, and in the other I2.
According to the cosine theorem, it is equal to:
If you rotate p by fi, you get i1 or i2, depending on which way you rotate. Next, the vector i1 or i2 must be added to a to obtain the intersection point
This method will work even if the center of one circle lies inside the other. But there the vector p will definitely have to be specified in the direction from a to b, which is what we did. If you build p based on another circle, then nothing will come of it
Well, in conclusion, one fact must be mentioned: if the circles touch, then it is easy to verify that P is the point of contact (this is true for both internal and external contact).
Here you can see the visualization (you need to click to launch it).
Problem 2: Intersection points
This method works, but instead of the rotation angle, you can calculate its cosine, and through it the sine, and then use them when rotating the vector. This will significantly simplify calculations by eliminating the code from trigonometric functions.
Class 7G, Z
Lesson topic: “The relative position of two circles”
Purpose: to know possible cases relative position of two circles; apply knowledge when solving problems.
Objectives: Educational: to facilitate the creation and consolidation in students of a visual representation of possible cases of the arrangement of two circles; students will be able to:
Establish a connection between the relative positions of circles, their radii and the distance between their centers;
Analyze a geometric design and mentally modify it,
Develop planimetric imagination.
Students will be able to apply theoretical knowledge to solving problems.
Lesson type: lesson introducing and consolidating new knowledge of the material.
Equipment: presentation for the lesson; compass, ruler, pencil and textbook for each student.
Tutorial: . “Geometry 7th grade”, Almaty “Atamura” 2012
During the classes.
Organizing time. Checking homework.
3. Updating of basic knowledge.
Repeat the definitions of circle, circle, radius, diameter, chord, distance from a point to a straight line.
1) 1) What cases of the location of a line and a circle do you know?
2) Which line is called tangent?
3) Which line is called a secant?
4) Theorem about diameter perpendicular to chord?
5) How is the tangent in relation to the radius of the circle?
6) Fill out the table (on cards).
- Students, under the guidance of the teacher, solve and analyze problems.
1) Line a is a tangent to a circle with center O. Point A is given on line a. The angle between the tangent and segment OA is 300. Find the length of segment OA if the radius is 2.5 m.
2) Determine the relative position of the line and the circle if:
- 1. R=16cm, d=12cm 2. R=5cm, d=4.2cm 3. R=7.2dm, d=3.7dm 4. R=8 cm, d=1.2dm 5. R=5 cm, d=50mm
a) a straight line and a circle do not have common points;
b) the line is tangent to the circle;
c) a straight line intersects a circle.
- d is the distance from the center of the circle to the straight line, R is the radius of the circle.
3) What can be said about the relative position of the line and the circle if the diameter of the circle is 10.3 cm and the distance from the center of the circle to the line is 4.15 cm; 2 dm; 103 mm; 5.15 cm, 1 dm 3 cm.
4) Given a circle with center O and point A. Where is point A located if the radius of the circle is 7 cm and the length of the segment OA is: a) 4 cm; b) 10 cm; c) 70 mm.
4. Together with the students, find out the topic of the lesson and formulate the goals of the lesson.
5. Introduction of new material.
Practical work in groups.
Construct 3 circles. For each circle, construct another circle so that 1) 2 circles do not intersect, 2) 2 circles touch, 3) two circles intersect. Find the radius of each circle and the distance between the centers of the circles, compare the results. What can be the conclusion?
2) Summarize and write down in a notebook the cases of the relative position of two circles.
The relative position of two circles on a plane.
The circles have no common points (do not intersect). (R1 and R2 are the radii of the circles)
If R1 + R2< d,
d – Distance between the centers of circles.
c) Circles have two common points. (intersect).
If R1 + R2 > d,
Question. Can two circles have three common points?
6. Consolidation of the studied material.
Find an error in the data or statement and correct it, justifying your opinion:
A) Two circles touch. Their radii are equal to R = 8 cm and r = 2 cm, the distance between the centers is d = 6.
B) Two circles have at least two points in common.
B) R = 4, r = 3, d = 5. Circles have no common points.
D) R = 8, r = 6, d = 4. The smaller circle is located inside the larger one.
D) Two circles cannot be positioned so that one is inside the other.
7. Lesson summary. What did you learn in the lesson? What pattern was established?
How can two circles be positioned? In what case do circles have one common point? What is the common point of two circles called? What touches do you know? When do circles intersect? What circles are called concentric?
Lesson topic: " The relative position of two circles on a plane.”
Target :
educational - mastering new knowledge about the relative position of two circles, preparing for test work
Developmental - development of computational skills, development of logical-structural thinking; developing skills in finding rational solutions and achieving final results; development cognitive activity and creative thinking .
Educational – formation of responsibility and consistency in students; development of cognitive and aesthetic qualities; formation information culture students.
Correctional - develop spatial thinking, memory, hand motor skills.
Lesson type: learning new things educational material, fastening.
Type of lesson: mixed lesson.
Teaching method: verbal, visual, practical.
Form of study: collective.
Means of education: board
DURING THE CLASSES:
1. Organizational stage
- greetings;
- checking preparedness for the lesson;
2.
Updating of basic knowledge.
What topics did we cover in previous lessons?
General form equations of a circle?
Perform orally:
Blitz Poll
3. Introduction of new material.
What figure do you think we will consider today... What if there are two of them??
How can they be located???
Children show with their hands (neighbors) how the circles can be arranged (physical education minute)
Well, what do you think we should consider today?? Today we should consider the relative position of two circles. And find out what the distance between the centers is depending on the location.
Lesson topic: « The relative position of two circles. Problem solving. »
1. Concentric circles
2. Disjoint circles
3.External touch
4. Intersecting circles
5. Internal touch
So let's conclude
4.Formation of skills and abilities
Find an error in the data or statement and correct it, justifying your opinion:
A) Two circles touch. Their radii are equal to R = 8 cm and r = 2 cm, the distance between the centers is d = 6.
B) Two circles have at least two points in common.
B) R = 4, r = 3, d = 5. Circles have no common points.
D) R = 8, r = 6, d = 4. The smaller circle is located inside the larger one.
D) Two circles cannot be positioned so that one is inside the other.
5. Consolidation of skills and abilities.
The circles touch externally. The radius of the smaller circle is 3 cm. The radius of the larger circle is 5 cm. What is the distance between the centers?
Solution: 3+5=8(cm)
The circles touch internally. The radius of the smaller circle is 3 cm. The radius of the larger circle is 5 cm. What is the distance between the centers of the circles?
Solution: 5-3=2(cm)
The circles touch internally. The distance between the centers of the circles is 2.5 cm. What are the radii of the circles?
answer: (5.5 cm and 3 cm), (6.5 cm and 4 cm), etc.
CHECKING COMPREHENSION
1) How can two circles be positioned?
2) In what case do circles have one common point?
3) What is the common point of two circles called?
4) What touches do you know?
5) When do the circles intersect?
6) What circles are called concentric?
Additional tasks on the topic: Vectors. Coordinate method "(if there is time left)
1)E(4;12),F(-4;-10), G(-2;6), H(4;-2) Find:
a) vector coordinatesEF, GH
b) vector lengthFG
c) coordinates of point O - the middleEF
point coordinatesW– middleGH
d) equation of a circle with diameterFG
e) equation of a lineFH
6. Homework
& 96 No. 1000. Which of these equations are equations of a circle. Find center and radius
7. Summing up the lesson (3 min.)
(give a qualitative assessment of the work of the class and individual students).
8. Reflection stage (2 minutes.)
(initiate student reflection on their emotional state, their activities, interaction with the teacher and classmates using drawings)